1. Introduction
Human life is stable and immovable due to energy. The development and progress of energy are necessary for a better life. The conventional sources of energy are depleted and cause environmental exacerbation, so the dependence on energy from renewable energy sources is inevitable as they are clean, have no environmental problems, exist in large quantities and provide energy with high capability [1,2,3,4,5]. One of the most important renewable energy sources is solar energy, where solar irradiation can be transformed effectively into electrical energy via photovoltaic (PV) cells/modules and may directly supply electric loads or be stored in batteries or other storage devices [6,7].
Several advanced applications have been introduced based on PV electricity, such as feeding the required power for satellite communication [8], greenhouse cooling and heating [9], water pumping for agriculture [10,11,12], supplying electronic devices and indoor lighting [13,14], etc. The PV characteristics can be analyzed with power–voltage (P–V) and current–voltage (I–V) curves. These curves are dependent on several parameters, such as incident solar irradiance, ambient temperature, and the investigated equivalent circuit of the PV model [15,16,17,18]. The PV characteristics depend on different unknown parameters due to a lack of data from the PV manufacturing datasheet [19]. Improving and analyzing the performance of PV cells/modules is imperative due to their widespread applications, which require optimal extraction of the unknown parameters. These parameters are changed according to the investigated PV models which can be a single diode model (SDM), double diode model (DDM), and three diode model (TDM). Consequently, the number of unknown parameters are five, seven and nine for the SDM, DDM, and TDM, respectively.
These parameters are estimated in three ways: iterative methods, machine learning, and meta-heuristic optimization algorithms [20,21,22,23,24,25,26]. The iterative methods have been applied to estimate the PV parameters in [27,28,29,30], such as Lambert W function [27], linear least squares [28], maximum likelihood-based Newton–Raphson [29], and Gauss–Seidel [30]. On the other hand, several researchers made an assumption or neglected some parameters to reduce the number of variables required to be extracted.
Lately, various optimization techniques have been carried out in the extraction of PV parameters, such as the elephant herd algorithm [31], multiple learning backtracking search algorithm [32], gray wolf optimizer, cuckoo search algorithm [33], opposition-based sine cosine approach with local search [34], logistic chaotic JAYA algorithm [35], moth–flame algorithm (MFA), orthogonal Nelder–Mead MFA [36], and improved teaching–learning-based optimization (TBLO) algorithm [37]. In [38], the MFA has been utilized for the three diode PV model considering the ideality factors for the second and third diode as added control variables. In [39], these parameters have been estimated with an interval branch and bound global optimization algorithm. In [40], simplified TBLO has been applied to estimate the parameters in a TDM. In addition, parameter extraction has been prepared by an improved version of the whale optimization algorithm [41] and chaotic improved artificial bee colony (CIABC) [42]. There is no doubt that the accuracy of the behavior of PVs is based on the estimated parameters, so the optimization techniques need further development to achieve high accuracy of these parameters. Additionally, in [43], another optimization method called forensic optimizer was developed for finding the optimal parameters of various solar cells. In [44], the gradient based optimizer was developed for three diode models.
As seen in the literature, incredible work has been performed in the extraction of the optimal PV model parameters. A global solution has not been accomplished as the randomization process is a property of all optimization search algorithms. Among of the previous optimization methods, a new optimization method called turbulent flow of water-based optimization (TFWO) [45] is developed for finding the parameters of three models of PV cells. Several new optimization techniques, such as the marine predators algorithm (MPA), equilibrium optimization (EO), and manta ray foraging optimization (MRFO), are used to compare the results of the proposed algorithm with the same dataset. Statistical analysis is used to analyze the performance of the proposed optimization algorithm. The P–V and I–V curves are simulated for the value of the estimated parameter that makes the simulated data very close to the experimental data.
The organization of this paper is as follows: Section 2 explains the analysis of the objective function to be handled in the problem formulation. Section 3 contains the details of the proposed TFWO algorithm. Section 4 analyzes the results of the studied cases, while the conclusion is drawn in Section 5.
2. Problem Formulation and Objective Function
Three models of PVs are analyzed in this section, SDM, DDM, and TDM [44], to be formulated in the objective function.
2.1. Analysis of SDM
Figure 1 explains the SDM equivalent circuit of the PV solar cell. The mathematical equations to calculate the output current of the SDM can be formulated as follows:
(1)
(2)
where I is the current output from the solar cell SDM, is the photogenerated current, is the current due to leakage in the PN junction, is the dark saturation current of the SDM, is the shunt resistance, is the series resistance, is the diode ideality factor, K is Boltzmann’s constant, q is the charge of the electron, and is the cell temperature.According to the previous mathematical formula, the five unknown parameters required to estimate the SDM are .
2.2. Analysis of DDM
Figure 2 explains the DDM equivalent circuit of the PV solar cell. The mathematical equations to compute the output current of the DDM are as follows:
(3)
(4)
where is the dark saturation current of the second diode in the DDM, is the ideality factor of the second diode. In this model, seven parameters should be estimated, which are .2.3. Analysis of TDM
Figure 3 illustrates the TDM equivalent circuit related to the PV solar cell. The mathematical equations to compute the output current of the TDM are as follows:
(5)
(6)
where is the dark saturation current of the third diode in the TDM, is the ideality factor of the third diode. In this model, nine parameters should be estimated, which are .2.4. Estimated Objective Function
Minimizing the root mean square error (RMSE) of the PV characteristics between the estimated parameters and the experimental results is an important objective function to be considered. Therefore, the decision variables (X) are extracted in each run of the optimizer. The mathematical formula to compute RMSE can be formulated as follows:
(7)
(8)
where is the experimental current, N is the reading data number, V is the experimental voltage, I is the estimated current, and X is the decision variables that are calculated as follows:For SDM, .
For DDM,
For TDM, .
3. Proposed Turbulent Flow of Water-Based Optimization Algorithm
The turbulent flow of water-based optimization algorithm (TFWOA), which was presented by Mojtaba Ghasemi et al. [45], is inspired by the principle of irregular fluctuations of water turbulent flow. In this type of turbulent flow, the magnitude and direction speed are continuously changing in a circular form. Then, the water flows downwards in a spiral path. In this algorithm, a whirlpool represents a random behavior of nature that can occur in seas, oceans or rivers. The center of the whirlpool is considered a sucking hole, and it pulls the particles across it towards the middle. To illustrate, the whirlpool uses centripetal force on them, which involves a volume of moving water created by the ocean tide. Centripetal force is characterized as a force that is employed in a circular path on a moving object, and its direction is in the direction of the center of the motion pathway of the object and perpendicular to it. The centripetal force shifts the moving pathway of the object without changing the velocity. Firstly, the initial population of the algorithm ( members) (comprising ) is split into an equal rate between groups which represent the whirlpool sets. Secondly, the strongest member of each whirlpool set (the member with better objective function values) is considered as the whirlpool that pulls the objects.
Every whirlpool () behaves as a sucking well and has a tendency to unify the locations of objects inside its set () with its central position through applying a centripetal force on them and pushing them into its well. Thus, the whirlpool and the local position on combines the object position () with itself (). However, other whirlpools produce some deviations () because of the distance between them () and their objective values () as well. Accordingly, the new position of the object becomes .. and the objects () move with their special angle () across their whirlpool’s center and move toward it. Hence, this angle in each iteration is changing according to Equation (9):
(9)
To model and calculate the farthest and nearest whirlpools (), Equation (10) depicts the whirlpools with the least weighed distance from all objects, and then is calculated using Equation (11). Equation (12) is used to update the position of the particle.
(10)
(11)
(12)
where and manifest the whirlpools with the minimum and maximum of , respectively, while characterizes the object’s angle.Centrifugal force () sometimes overcomes the centripetal force of the whirlpool and randomly transfers the object to a new location. The centrifugal force is modeled as illustrated in Equation (13), which randomly occurs in one dimension of the decision variables. To attain this, the centrifugal force is calculated according to the angle between the whirlpool and object, as manifested in Equation (13), and if this force is greater than a random value in the range [0,1], the centrifugal action is performed for a randomly selected dimension, as shown in Equation (14). This phenomenon is formulated mathematically as:
(13)
(14)
The whirlpools interact with and displace each other. This phenomenon can be modeled in the same way as the impacts of whirlpools on the objects, where every whirlpool has a tendency to pull other whirlpools and apply the centripetal force on them. The nearest whirlpool can be mathematically represented based on the minimum amount and its objective function, as illustrated in Equation (15). Then, the whirlpool’s position can be updated according to Equations (16) and (17).
(15)
(16)
(17)
where represents the whirlpool hole angle value.Eventually, when the strongest member has more strength among the new members of the whirlpool set, which means that the value of the objective function is less than its corresponding whirlpool, it is chosen as a new whirlpool for the next iteration. The flowchart of the TFWOA is depicted in Figure 4.
4. Simulation Results and Discussion
This section presents the application and analysis of the proposed TFWO algorithm for extracting the optimal values of the parameters of various PV models. Real data of a 55 mm diameter commercial R.T.C. France solar cell [7,44] and experimental data of a KC200GT module [46] are considered. The considered boundaries of the parameters are explained in Table 1.
4.1. Compared Algorithms
Several optimization algorithms are employed and compared to the proposed TFWO (Turbulent Flow of Water Optimizer) for the same purpose. These algorithms are the backtracking search optimization algorithm (BSA) [47], gray wolf optimizer (GWO) [48], crow search optimization algorithm (CSO) [49], equilibrium optimizer (EO) [50], marine predators algorithm (MPA) [51], Bernstein–Levy search differential evolution algorithm (BSDE) [52] and manta ray foraging optimization (MRFO) [53]. The BSA, GWO and CSO have different successive applications, while the EO, MPA, BSDE, and MRFO are very recent algorithms from 2020. Table 2 represents examples of their recent applications.
All these algorithms have the merit of utilizing adaptive internal control parameters. For all algorithms, the population size is specified as 100, where the maximum number of iterations is taken as 1000 and 2000 for an R.T.C. France solar cell and KC200GT module, respectively. The compared algorithms in Table 3 are employed for optimal extraction of the PV parameters with the SDM, DDM, and TDM. The convergence performance, robustness, and accuracy for all algorithms used in this work are found based on 30 separate runs for each algorithm.
4.2. Statistical Analysis for R.T.C. France Solar Cell
4.2.1. Single Diode Model
Table 3 provides the optimal values of the control variables related to the best run for the compared algorithms. As shown, TFWO obtains the minimum RMSE of 0.000986022 compared to the others. Based on TFWO, the photo-generated current is 0.760775529 A; the dark saturation current of the SDM is 0.323 μA; the diode ideality factor is 1.481183723; the series resistance is 0.036377085 Ω; the shunt resistance is 53.71858096 Ω. Figure 5 describes the convergence rates of the algorithms and shows that the capability of the proposed TFWO in finding the minimum RMSE is the fastest.
Based on the extracted PV parameters using the TFWO, Figure 6 describes the I–V and P–V characteristic curves in comparison to the experimental data. This figure illustrates the great similarity between the extracted curves based on TFWO and the experimental results. Figure 7 shows this capability, where the error for each value of current and power is shown between the simulated and experimental data to measure the quality of the result.
Table 4 records the minimum, maximum, mean, and standard deviation of the RMSE for the SDM. This table declares that TFWO presents the highest robustness characteristics. It gives the lowest values of the minimum, maximum, mean, and standard deviation of the RMSE, at 0.000986022, 0.000986205, 0.00098603, and 3.35307 × 10−8, respectively. Meanwhile, the second-best RMSE (0.000986023) is achieved by the BSDE, followed by MRFO, EO, CSO, MPA, BSA, and GWO. Figure 8 displays the RMS values of the 30 runs for the R.T.C. France SDM. This figure shows the significant robustness feature of the proposed TFWO since all the acquired values of the RMSE based on TFWO are the lowest values compared with the other methods.
4.2.2. Double Diode Model
The proposed TFWO and the compared algorithms are applied for this model. Table 5 provides the optimal values of the control variables related to the best run, while Figure 9 describes their convergence rates. From both, it can be observed that the best RMSE value (0.000982723) is achieved by the TFWO algorithm, while the second-best RMSE (0.000983378) is achieved by MRFO, followed by CSO, EO, BSDE, BSA, GWO, and MPA.
Based on the extracted PV parameters using the TFWO, Figure 10 describes the I–V and P–V characteristic curves in comparison to the experimental data, while Figure 11 displays the related errors. From both figures, the coincidence of the simulated data based on TFWO with the experimental data is very high.
Table 6 records the minimum, maximum, mean, and standard deviation of the RMSE for the DDM. This table declares that TFWO presents the highest robustness characteristics. It gives the lowest values of the minimum, maximum, mean, and standard deviation of the RMSE as 0.000982723, 0.0012, 0.00099392, and 3.9352 × 10−5, respectively. Figure 12 displays the RMSE values of the 30 runs for the R.T.C. France DDM. The acquired values of the RMSE based on the proposed TFWO are lower than their comparable values based on the others.
4.2.3. Three Diode Model
For this model, Table 7 provides the optimal values of the control variables related to the best run of the proposed TFWO and the compared algorithms, while Figure 13 describes their convergence rates. From both, it can be observed that the best RMSE value (0.000983646) is achieved by the TFWO algorithm, while the second-best RMSE (0.000984242) is achieved by CSO, followed by MRFO, EO, BSA, MPA, BSDE, and GWO. Figure 14 describes the I–V and P–V characteristic curves in comparison with the experimental data, while Figure 15 displays the related errors. From both figures, the coincidence of the simulated data based on TFWO with the experimental data is very high.
For the 30 runs, the minimum, maximum, mean, and standard deviation of the RMSE are tabulated in Table 8. As shown, the proposed TFWO gives the lowest values of the minimum, maximum, mean, and standard deviation as 0.000983646, 0.00102314, 0.000987683, and 7.32713 × 10−6, respectively. Figure 16 displays the RMSE values of the 30 runs for the R.T.C. France TDM, which demonstrate the efficacy of the proposed TFWO in finding the minimum RMSE values compared to the others.
4.3. Statistical Analysis for KC200GT Solar Module
4.3.1. Single Diode Model
The comparison of the results for the SDM is explained in Table 9; this table includes the best RMSE and the parameters extracted from each algorithm. From Table 9, it can be observed that the best RMSE value (0.000636657) is achieved by the TFWO algorithm, while the second-best RMSE (0.002888472) is achieved by EO, followed by MRFO, BSDE, BSA, MPA, CSO, and GWO. Based on TFWO, the photo-generated current is 8.216747428 A; the dark saturation current of the SDM is 0.0262486 μA; the diode ideality factor is 1.212957711; the series resistance is 0.004825464 Ω; the shunt resistance is 6.284632281 Ω. Figure 17 describes the convergence rates of the algorithms which show that the capability of the proposed TFWO in finding the minimum RMSE is the fastest. Added to that, the P–V and I–V curves for the SDM based on the estimated data from TFWO at the best RMSE are explained in Figure 18, which illustrates the high coincidence of the simulated with the experimental data.
4.3.2. Double Diode Model
For this model, Table 10 shows the optimal values of the control variables related to the best run of the compared algorithms, while Figure 19 illustrates their convergence rates. From both, it can be observed that the best RMSE value (0.000464919) is achieved by the TFWO algorithm, while the second-best RMSE (0.002599915) is achieved by EO, followed by CSO, MRFO, GWO, BSA, BSDE, and MPA. Figure 20 describes the I–V and P–V characteristic curves in comparison to the experimental data.
4.3.3. Three Diode Model
For this model, Table 11 and Figure 21 show the optimal values of the control variables of the compared algorithms and their convergence rates, respectively. From both, the best RMSE value (0.000379678) is achieved by the proposed TFWO. The P–V and I–V curves for the TDM based on the estimated data from TFWO at the best RMSE are explained in Figure 22, whilst the error for each value of current and power between the simulated and experimental data is found to measure the quality of the result, as shown in Figure 23. From both, the coincidence of the simulated data based on TFWO with the experimental data is very high.
4.3.4. Statistical Analysis for KC200GT Models
For the KC200GT module, the robustness accuracy for all algorithms is evaluated for the SDM, DDM, and TDM. Table 12 records the minimum, maximum, mean, and standard deviation of the RMSE for the DDM. This table declares that TFWO presents the highest robustness characteristics. It gives the lowest values of the minimum, maximum, mean, and standard deviation of the RMSE for the three PV models.
5. Conclusions
In this paper, a new application has been carried out for a new optimation algorithm called turbulent flow of water-based optimization (TFWO) for the parameter extraction of three models of PV cells. These applications are implemented on the real data of a 55 mm diameter commercial R.T.C. France solar cell and experimental data of a KC200GT module. An assessment study comparing several recent optimization techniques is employed to show the capability of the proposed TFWO algorithm. The comparative study is carried out for the same dataset and for the same computation burden. Statistical analysis is used to analyze the performance of the proposed TFWO algorithm. The high closeness between the estimated P–V and I–V curves is achieved by the proposed TFWO compared with the experimental data as well as the competitive optimization algorithms. Added to that, the proposed method has a robust performance as well as good convergence rates for all tested cases.
In future work, various environmental impacts, such as temperature, moisture, and noise, as well as the unidentifiability of parameters concept presented in [70,71,72,73], are suggested to be considered for different models as an extension of this work. Another direction is the development of solution methods with a multi-objective framework that combines the closeness of parameters and maximum benefits for power system operators.
Author Contributions
All authors have contributed to the preparation of this manuscript. M.S., A.M.S. and A.R.G. designed the strategy, studied the data, and wrote the manuscript. M.M.F.D. and K.M. revised the manuscript and investigated the optimization methodology. Finally, M.L. and R.A.E.-S. reviewed, edited, and supported different improvements to the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by the Department of Electrical Engineering and Automation, School of Electrical Engineering, Aalto University, Espoo, Finland.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
The data presented in this study are available on request from the corresponding author.
Conflicts of Interest
The authors declare no conflict of interest.
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Figures and Tables
Figure 4. Flowchart of the turbulent flow of water-based optimization algorithm (TFWOA).
Figure 6. Characteristic curves for R.T.C. France SDM based on parameters extracted from TFWO: (a) Current–voltage (I–V) ch/s and (b) power–voltage (P–V) ch/s.
Figure 7. Error values for R.T.C. France SDM based on parameters extracted from TFWO: (a) Current error values and (b) power error values.
Figure 10. Characteristic curves for R.T.C. France DDM based on TFWO: (a) I–V ch/s and (b) P–V ch/s.
Figure 11. Error values for R.T.C. France DDM based on TFWO: (a) Current error values and (b) power error values.
Figure 14. Characteristic curves for R.T.C. France TDM based on TFWO: (a) I–V ch/s and (b) P–V ch/s.
Figure 15. Error values for R.T.C. France TDM based on TFWO: (a) Current error values and (b) power error values.
Figure 15. Error values for R.T.C. France TDM based on TFWO: (a) Current error values and (b) power error values.
Figure 18. Characteristic curves for KC200GT SDM based on TFWO: (a) I–V ch/s and (b) P–V ch/s.
Figure 20. Characteristic curves for KC200GT DDM based on TFWO: (a) I–V ch/s and (b) P–V ch/s.
Figure 22. Characteristic curves for KC200GT TDM based on TFWO: (a) I–V ch/s and (b) P–V ch/s.
Figure 23. Error values for KC200GT TDM based on TFWO: (a) Current error values and (b) power error values.
Figure 23. Error values for KC200GT TDM based on TFWO: (a) Current error values and (b) power error values.
The extracted parameters boundaries of test solar cells and modules.
R.T.C. France Solar Cell [7] | KC200GT Module [7] | |||
---|---|---|---|---|
Parameters | Lower Bound | Upper Bound | Lower Bound | Upper Bound |
0 | 1 | 0 | 9 | |
(μA) | 0 | 1 | 0 | 1 |
0 | 0.5 | 0 | 0.5 | |
0 | 100 | 0 | 100 | |
1 | 2 | 1 | 2 |
Several recent applications of the compared algorithms.
Algorithm | Published Year | Recent Applications |
---|---|---|
BSA [47] | 2013 | Reconfiguration in distribution networks (2020) [54], reactive power dispatch (2018) [55], parameter optimization of the support vector machine (2020) [56]. |
GWO [48] | 2014 | Coordination of VAR compensators and distributed energy resources (2020) [57], allocation of distributed generation in power systems (2020) [58], energy management, and battery size optimization (2020) [59]. |
CSO [49] | 2016 | Short-term wind speed forecasting (2020) [60], capacitor allocation in distribution networks (2017) [61], emission economic dispatch [62] |
EO [50] | 2020 | Multi-thresholding image segmentation problems [63], operation of hybrid AC/DC grids (2020) [64]. |
MPA [51] | 2020 | Large-scale photovoltaic array reconfiguration (2020) [65], task scheduling in IoT-based fog computing applications (2020) [66]. |
BSDE [52] | 2021 | Not applicable yet. |
MRFO [53] | 2020 | Fuel cell exergy analysis (2020) [67], optimal power flow (2020) [68], maximum power point (2020) [69]. |
TFWO [45] | 2021 | Not applicable yet. |
The parameters extracted for R.T.C. France SDM at the best root mean square error (RMSE).
Algorithm | Iph (A) | Id1 (A) | a 1 | Rs (Ω) | Rsh (Ω) | RMSE |
---|---|---|---|---|---|---|
TFWO | 0.760775529 | 3.23 × 10−7 | 1.481183723 | 0.036377085 | 53.71858096 | 0.000986022 |
MRFO | 0.760778817 | 3.22884 × 10−7 | 1.481141648 | 0.036380748 | 53.67819867 | 0.000986034 |
BSDE | 0.760773529 | 3.23008 × 10−7 | 1.481179386 | 0.036378015 | 53.74364455 | 0.000986023 |
MPA | 0.760846832 | 3.22991 × 10−7 | 1.48119268 | 0.036361364 | 52.76698061 | 0.000987369 |
EO | 0.76077794 | 3.22162 × 10−7 | 1.480915424 | 0.036387935 | 53.64156933 | 0.000986035 |
CSO | 0.760757142 | 3.24211 × 10−7 | 1.481546607 | 0.036366384 | 54.0990705 | 0.000986181 |
GWO | 0.760695583 | 3.58429 × 10−7 | 1.491737687 | 0.035974121 | 57.26269608 | 0.001008231 |
BSA | 0.760850914 | 3.11696 × 10−7 | 1.477614787 | 0.036510097 | 51.96067738 | 0.000989471 |
Statistical analysis of RMSE for R.T.C. France SDM.
Algorithm | RMSE | |||
---|---|---|---|---|
Min. | Max. | Mean | SD | |
TFWO | 0.00098602 | 0.00098620 | 0.00098603 | 3.353 × 10−8 |
MRFO | 0.00098603 | 0.00105788 | 0.00100505 | 2.143 × 10−5 |
BSDE | 0.00098602 | 0.00103025 | 0.00099520 | 1.056 × 10−5 |
MPA | 0.00098736 | 0.00481175 | 0.00217485 | 0.00065237 |
EO | 0.00098603 | 0.00105604 | 0.00100209 | 1.783 × 10−5 |
CSO | 0.00098618 | 0.00130296 | 0.00105888 | 8.095 × 10−5 |
GWO | 0.00100823 | 0.03816637 | 0.00637283 | 0.0112567 |
BSA | 0.000989471 | 0.001161862 | 0.001037488 | 4.42885 × 10−5 |
The parameters extracted for R.T.C. France DDM.
Algorithm | Iph (A) | Rs (Ω) | Rsh (Ω) | RMSE | Id1 (A) | a 1 | Id2 (A) | a 2 |
---|---|---|---|---|---|---|---|---|
TFWO | 0.760782016 | 0.036839463 | 55.91920478 | 0.000982723 | 2.06 × 10−7 | 1.443289469 | 9.24 × 10−7 | 2 |
MRFO | 0.760743575 | 0.036597626 | 54.95169271 | 0.000983378 | 4.37429 × 10−7 | 1.998364786 | 2.62887 × 10−7 | 1.463671024 |
BSDE | 0.760782257 | 0.036991096 | 54.62889674 | 0.000989247 | 1.38431 × 10−7 | 1.416972632 | 5.71114 × 10−7 | 1.764756216 |
MPA | 0.760918727 | 0.037865706 | 53.18011281 | 0.001026823 | 7.66125 × 10−8 | 1.36857525 | 9.99997 × 10−7 | 1.815337209 |
EO | 0.760741801 | 0.036329661 | 54.62831228 | 0.000986861 | 3.06281 × 10−7 | 1.492418793 | 2.85646 × 10−8 | 1.428995768 |
CSO | 0.760756875 | 0.03652498 | 54.63222775 | 0.000983888 | 3.22867 × 10−7 | 1.992580518 | 2.7755× 10−7 | 1.46831668 |
GWO | 0.760583028 | 0.036533827 | 58.81767959 | 0.001003603 | 3.2814 × 10−7 | 1.563347542 | 8.25411× 10−8 | 1.41082037 |
BSA | 0.760980002 | 0.036723119 | 53.23192348 | 0.000993668 | 2.64414 × 10−7 | 1.705891588 | 1.99393 × 10−7 | 1.446025602 |
Statistical analysis of RMSE for R.T.C. France DDM.
Algorithm | RMSE | |||
---|---|---|---|---|
Min. | Max. | Mean | SD | |
TFWO | 0.000982723 | 0.0012 | 0.00099392 | 3.9352 × 10−5 |
MRFO | 0.000983378 | 0.001353061 | 0.001077661 | 8.45223 × 10−5 |
BSDE | 0.000989247 | 0.001492072 | 0.001113348 | 0.000112212 |
MPA | 0.001026823 | 0.002869201 | 0.001779704 | 0.000616954 |
EO | 0.000986861 | 0.001256857 | 0.001033158 | 6.33746 × 10−5 |
CSO | 0.000983888 | 0.001428127 | 0.00113901 | 0.000155378 |
GWO | 0.001003603 | 0.038150899 | 0.00640054 | 0.011254562 |
BSA | 0.000993668 | 0.001214824 | 0.001080621 | 5.45125 × 10−5 |
The parameters extracted for R.T.C. France TDM.
Algorithm | BSA | GWO | CSO | EO | MPA | BSDE | MRFO | TFWO |
---|---|---|---|---|---|---|---|---|
Iph (A) | 0.76088788 | 0.761840018 | 0.760767839 | 0.760733925 | 0.760665312 | 0.76060128 | 0.760721516 | 0.7608 |
Is1 (A) | 6.11525 × 10−8 | 6.26386 × 10−7 | 8.65078 × 10−7 | 2.29078 × 10−7 | 2.60174 × 10−15 | 1.33125 × 10−7 | 2.6918 × 10−7 | 0 |
a 1 | 1.665282347 | 1.972178035 | 1.992247826 | 1.945636832 | 1.025249938 | 1.715044852 | 1.880941278 | 1 |
Rs (Ω) | 0.036740001 | 0.036238306 | 0.036859716 | 0.036427424 | 0.037130986 | 0.036693181 | 0.036566016 | 0.0367 |
Rsh (Ω) | 53.18712346 | 43.25339883 | 54.98736983 | 55.52914763 | 59.57973022 | 60.17938354 | 55.20751535 | 55.2261 |
Is2 (A) | 8.13561 × 10−8 | 7.69448 × 10−9 | 4.64418 × 10−11 | 9.51489 × 10−8 | 6.85783 × 10−7 | 2.119 × 10−7 | 7.27156 × 10−8 | 2.39243 × 10−7 |
a2 | 1.951596911 | 1.982906016 | 1.583874031 | 1.981076476 | 1.670531807 | 1.449412901 | 1.755268871 | 1.4558 |
Is3 (A) | 2.62168 × 10−7 | 2.41707 × 10−7 | 2.06159× 10−7 | 2.78562 × 10−7 | 4.54209 × 10−8 | 4.27168 × 10−7 | 2.41083 × 10−7 | 6.38605 × 10−7 |
a3 | 1.464894557 | 1.457809556 | 1.443266003 | 1.469223538 | 1.348012837 | 1.942277098 | 1.458171437 | 2 |
RMSE | 0.001002321 | 0.001293402 | 0.000984242 | 0.000985451 | 0.001002377 | 0.001029117 | 0.000984843 | 0.000983646 |
Statistical analysis of RMSE for R.T.C. France TDM.
Algorithm | RMSE | |||
---|---|---|---|---|
Min. | Max. | Mean | SD | |
TFWO | 0.000983646 | 0.00102314 | 0.000987683 | 7.32713 × 10−6 |
MRFO | 0.000984843 | 0.001492643 | 0.001164256 | 0.000129441 |
BSDE | 0.001029117 | 0.002051955 | 0.001320873 | 0.000243075 |
MPA | 0.001002377 | 0.005305369 | 0.002200116 | 0.000900812 |
EO | 0.000985451 | 0.001393105 | 0.001131243 | 0.00011292 |
CSO | 0.000984242 | 0.00191729 | 0.001164341 | 0.000184491 |
GWO | 0.001293402 | 0.033393772 | 0.006435657 | 0.01033541 |
BSA | 0.001002321 | 0.001567976 | 0.001189651 | 0.000119982 |
Extracted parameters for KC200GT SDM.
Algorithm | Iph (A) | Is1 (A) | a 1 | Rs (Ω) | Rsh (Ω) | RMSE |
---|---|---|---|---|---|---|
TFWO | 8.216747428 | 2.62486 × 10−8 | 1.212957711 | 0.004825464 | 6.284632281 | 0.000636657 |
MRFO | 8.212405132 | 3.36662 × 10−8 | 1.228520397 | 0.004754881 | 7.037075568 | 0.003374264 |
BSDE | 8.210553583 | 3.43101 × 10−8 | 1.229705769 | 0.004756865 | 7.555908952 | 0.003467884 |
MPA | 8.184927 | 7.94459 × 10−8 | 1.285180059 | 0.004537611 | 92.14823504 | 0.0148696 |
EO | 8.209152899 | 2.85259 × 10−8 | 1.218067754 | 0.004814539 | 7.714703106 | 0.002888472 |
CSO | 8.188955905 | 8.18358 × 10−8 | 1.287282057 | 0.004540479 | 87.91105559 | 0.015480743 |
GWO | 8.193721562 | 1.72203 × 10−7 | 1.341187392 | 0.004264421 | 84.34172349 | 0.023476598 |
BSA | 8.187828492 | 4.39672 × 10−8 | 1.245523356 | 0.004706406 | 17.16016059 | 0.009775873 |
Extracted parameters for KC200GT DDM.
Algorithm | Iph (A) | Rs (Ω) | Rsh (Ω) | Is1 (A) | a 1 | Is2 (A) | a2 | RMSE |
---|---|---|---|---|---|---|---|---|
TFWO | 8.215931265 | 0.00490447 | 6.55275986 | 9.75 × 10−11 | 1 | 4.58 × 10−8 | 1.266697565 | 0.000464919 |
MRFO | 8.207554293 | 0.004729 | 7.962198358 | 1.30925 × 10−7 | 1.956231371 | 3.89385 × 10−8 | 1.237993335 | 0.008229492 |
BSDE | 8.199742079 | 0.004618981 | 11.00371597 | 1.70333 × 10−7 | 1.898851719 | 5.22564 × 10−8 | 1.257319714 | 0.009849963 |
MPA | 8.184775806 | 0.005037849 | 96.10264033 | 8.62345 × 10−7 | 1.581206361 | 4.01866 × 10−10 | 1.017081239 | 0.01025436 |
EO | 8.210884382 | 0.004777302 | 7.422135219 | 9.02611 × 10−9 | 1.822712307 | 3.13628 × 10−8 | 1.224039636 | 0.002599915 |
CSO | 8.204148086 | 0.004890878 | 9.331329018 | 7.23319 × 10−8 | 1.304927843 | 1.27128 × 10−10 | 1.000421712 | 0.004212996 |
GWO | 8.188942442 | 0.004865207 | 20.87443954 | 7.54227 × 10−7 | 1.765240036 | 1.56333 × 10−8 | 1.185224915 | 0.009625309 |
BSA | 8.204090314 | 0.004601853 | 10.29800978 | 5.53 × 10−8 | 1.260653585 | 3.03837 × 10−8 | 1.998785758 | 0.009625725 |
Extracted parameters for KC200GT TDM.
Algorithm | BSA | GWO | CSO | MPA | EO | BSDE | MRFO | TFWO |
---|---|---|---|---|---|---|---|---|
Iph (A) | 8.20173508 | 8.194693695 | 8.181855948 | 8.17852875 | 8.197397535 | 8.202679685 | 8.196629725 | 8.216333065 |
Is1 (A) | 0.004614443 | 0.004525605 | 0.004692599 | 0.004752779 | 0.004683395 | 0.004733114 | 0.004684349 | 0.004855332 |
a 1 | 13.66542752 | 23.11163887 | 99.9201098 | 99.98707579 | 14.01948329 | 9.497022092 | 11.43921825 | 6.406246831 |
Rs (Ω) | 3.75184 × 10−8 | 8.7 × 10−9 | 9.49405 × 10−8 | 2.87148 × 10−7 | 3.86696 × 10−8 | 2.40491× 10−7 | 3.43285 × 10−7 | 1.65 × 10−14 |
Rsh (Ω) | 1.238796687 | 1.590792134 | 1.481550006 | 1.983137731 | 1.238348021 | 1.797831823 | 1.89039067 | 1.00002872 |
Is2 (A) | 1.6158 × 10−7 | 6.62556 × 10−9 | 1.99524 × 10−8 | 3.79051 × 10−8 | 7.61616 × 10−7 | 3.2756 × 10−8 | 3.85165 × 10−8 | 2.04 × 10−9 |
a2 | 1.775790984 | 1.295053844 | 1.213009208 | 1.236293271 | 1.991605757 | 1.228057245 | 1.238261076 | 1.11890891 |
Is3 (A) | 4.33175 × 10−7 | 7.22637 × 10−8 | 1.60734 × 10−8 | 1.20422 × 10−7 | 2.11766 × 10−7 | 1.75316 × 10−7 | 7.55869 × 10−8 | 3.78866 × 10−8 |
a 3 | 1.743389601 | 1.284146864 | 1.310004248 | 1.967323029 | 1.958780659 | 1.917122756 | 1.748171665 | 1.270101351 |
RMSE | 0.011035788 | 0.013924443 | 0.013060563 | 0.013504282 | 0.008423459 | 0.006771142 | 0.008878327 | 0.000379678 |
Statistical analysis of RMSE for KC200GT module with SDM, DDM, and TDM.
Model | Algorithm | RMSE | |||
---|---|---|---|---|---|
Min. | Max. | Mean | SD | ||
SDM | TFWO | 0.000636657 | 0.000776307 | 0.000643757 | 2.76367 × 10−5 |
MRFO | 0.003374264 | 0.015283988 | 0.01143509 | 0.003320893 | |
BSDE | 0.003467884 | 0.014320685 | 0.010188693 | 0.002289554 | |
MPA | 0.0148696 | 0.048448767 | 0.039118106 | 0.010156791 | |
EO | 0.002888472 | 0.01320854 | 0.009771334 | 0.002376063 | |
CSO | 0.015480743 | 0.023739498 | 0.019620651 | 0.002077078 | |
GWO | 0.023476598 | 0.468609138 | 0.164042826 | 0.180451636 | |
BSA | 0.009775873 | 0.020577736 | 0.015024514 | 0.002330391 | |
DDM | TFWO | 0.000464919 | 0.003991719 | 0.000784157 | 0.000677807 |
MRFO | 0.008229492 | 0.017508428 | 0.013106997 | 0.002099435 | |
BSDE | 0.009849963 | 0.029252608 | 0.016694172 | 0.004448939 | |
MPA | 0.01025436 | 0.049871846 | 0.035790405 | 0.012557606 | |
EO | 0.002599915 | 0.013710246 | 0.009972209 | 0.002673846 | |
CSO | 0.004212996 | 0.025007328 | 0.017745851 | 0.004339108 | |
GWO | 0.009625309 | 0.467884375 | 0.149515185 | 0.179605814 | |
BSA | 0.009625725 | 0.026600837 | 0.017425267 | 0.003984149 | |
TDM | TFWO | 0.000379678 | 0.026665602 | 0.001706197 | 0.004771068 |
MRFO | 0.008878327 | 0.023401173 | 0.014709966 | 0.003787592 | |
BSDE | 0.006771142 | 0.032728388 | 0.019233922 | 0.006086967 | |
MPA | 0.013504282 | 0.051722136 | 0.039748254 | 0.012810733 | |
EO | 0.008423459 | 0.015285328 | 0.011790041 | 0.001841909 | |
CSO | 0.013060563 | 0.025146228 | 0.017635053 | 0.003334599 | |
GWO | 0.013924443 | 0.4172306 | 0.226455866 | 0.174785582 | |
BSA | 0.011035788 | 0.026603176 | 0.018267773 | 0.00416284 |
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Abstract
Recently, the use of diverse renewable energy resources has been intensively expanding due to their technical and environmental benefits. One of the important issues in the modeling and simulation of renewable energy resources is the extraction of the unknown parameters in photovoltaic models. In this regard, the parameters of three models of photovoltaic (PV) cells are extracted in this paper with a new optimization method called turbulent flow of water-based optimization (TFWO). The applications of the proposed TFWO algorithm for extracting the optimal values of the parameters for various PV models are implemented on the real data of a 55 mm diameter commercial R.T.C. France solar cell and experimental data of a KC200GT module. Further, an assessment study is employed to show the capability of the proposed TFWO algorithm compared with several recent optimization techniques such as the marine predators algorithm (MPA), equilibrium optimization (EO), and manta ray foraging optimization (MRFO). For a fair performance evaluation, the comparative study is carried out with the same dataset and the same computation burden for the different optimization algorithms. Statistical analysis is also used to analyze the performance of the proposed TFWO against the other optimization algorithms. The findings show a high closeness between the estimated power–voltage (P–V) and current–voltage (I–V) curves achieved by the proposed TFWO compared with the experimental data as well as the competitive optimization algorithms, thanks to the effectiveness of the developed TFWO solution mechanism.
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1 Department of Electrical Engineering, Faculty of Engineering, Fayoum University, 43518 Fayoum, Egypt;
2 Department of Electrical Engineering, Faculty of Engineering, Suez University, 41522 Suez, Egypt;
3 Department of Electrical Engineering, Faculty of Engineering, Kafrelshiekh University, 33516 Kafrelshiekh, Egypt;
4 Department of Electrical Engineering and Automation, School of Electrical Engineering, Aalto University, FI-00076 Espoo, Finland;
5 Department of Electrical Engineering and Automation, School of Electrical Engineering, Aalto University, FI-00076 Espoo, Finland;
6 Department of Electrical Engineering and Automation, School of Electrical Engineering, Aalto University, FI-00076 Espoo, Finland;